Question

13 mark for review
the function $h$ is defined by $h(\theta)=a\cos(b\theta)$ for constants $a$ and $b$. if $h$ has a period of $\pi$ and an amplitude of 4, what are the values of $a$ and $b$?
a $a=2$ and $b=\frac{1}{2}$
b $a=2$ and $b=2$
c $a=4$ and $b=\frac{1}{2}$
d $a=4$ and $b=2$

Explanation:

Step1: Identify amplitude from formula

For $h(\theta) = a\cos(b\theta)$, amplitude is $|a|$. Given amplitude = 4, so $a=4$.

Step2: Calculate b using period formula

Period of $\cos(b\theta)$ is $\frac{2\pi}{|b|}$. Given period = $\pi$, solve:
$$\pi = \frac{2\pi}{b}$$
$$b = \frac{2\pi}{\pi} = 2$$

Answer:

D. $a = 4$ and $b = 2$